Please provide help with these proofs. Let f:R->R be a function such that f(x+y)

Question

Please provide help with these proofs.
Let f:R->R be a function such that f(x+y) = f(x) + f(y)for all x,y in R. Assume that the limit of f=L as x approaches 0exists. Prove that L = 0, and then prove that f has a limit atevery point c in R. [Hint: First note that f(2x) = f(x) + f(x) =2f(x) for x in R. Also note that f(x) = f(x-c)+f(c) for x,c inR] Please provide help with these proofs.
Let f:R->R be a function such that f(x+y) = f(x) + f(y)for all x,y in R. Assume that the limit of f=L as x approaches 0exists. Prove that L = 0, and then prove that f has a limit atevery point c in R. [Hint: First note that f(2x) = f(x) + f(x) =2f(x) for x in R. Also note that f(x) = f(x-c)+f(c) for x,c inR]

Explanation / Answer

For the first part, note that f(1) is real. Consider the sequencea_n = f(1/n). Since the limit of f(x) as x tends to 0 exists and isL, we know that the sequence a_n converges to L as n tends toinfinity. But a_n = f(1/n) = f(1) / n (since n f(1/n) =f(1/n)+f(1/n) + .... n-fold + f(1/n) = f(n/n)=f(1). Thus a_nconverges to 0. Therefore L=0. To see the second claim, let c in R be given. As noted,f(c)=f(x-c)+f(x). Thus 0

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